Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470867 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,29,35)(16,30,31,38)(17,22,18,23)(20,27,36,28)(21,32,37,33)(26,34,40,44)(39,42,43,41);(5,7)(6,9)(8,11)(10,12)(13,15,18,27,19,29,17,28)(14,20,22,35,24,36,23,25)(16,26,33,41,31,40,32,42)(21,34,30,43,37,44,38,39);(2,4)(7,11)(9,12)(13,16,17,32,19,31,18,33)(14,21,23,38,24,37,22,30)(15,26,28,42,29,40,27,41)(20,34,25,39,36,44,35,43);(7,11)(9,12)(16,31)(21,37)(26,40)(30,38)(32,33)(34,44)(39,43)(41,42);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,31,33)(20,25,36,35)(21,30,37,38)(26,41,40,42)(34,39,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378067);